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Bisexual branching diffusions

Part of: Stochastic analysis Stochastic processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Leonid Mytnik  and
Robert J. Adler
Leonid Mytnik*
Affiliation:
Technion-Israel Institute of Technology
Robert J. Adler*
Affiliation:
Technion-Israel Institute of Technology
*
* Postal address: Faculty of Industrial Engineering and Management, Technion-Israel Institute of Technology, Haifa 32000, Israel.
* Postal address: Faculty of Industrial Engineering and Management, Technion-Israel Institute of Technology, Haifa 32000, Israel.
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Abstract

We study the limiting behaviour of large systems of two types of Brownian particles undergoing bisexual branching. Particles of each type generate individuals of both types, and the respective branching law is asymptotically critical for the two-dimensional system, while being subcritical for each individual population.

The main result of the paper is that the limiting behaviour of suitably scaled sums and differences of the two populations is given by a pair of measure and distribution valued processes which, together, determine the limit behaviours of the individual populations.

Our proofs are based on the martingale problem approach to general state space processes. The fact that our limit involves both measure and distribution valued processes requires the development of some new methodologies of independent interest.

Keywords

BRANCHING PARTICLE SYSTEMSBISEXUAL BRANCHINGSUPERPROCESSESWEAK CONVERGENCEMARTINGALE PROBLEM

MSC classification

Primary: 60G57: Random measures 60G20: Generalized stochastic processes
Secondary: 60F17: Functional limit theorems; invariance principles 60H15: Stochastic partial differential equations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 4 , December 1995 , pp. 980 - 1018
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

Research supported in part by US-Israel Binational Science Foundation (92-074), and Israel Academy of Sciences (025-93).

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